Graph polynomial functions using zeros, factorizations, and end behavior.
Build the least-degree polynomial from graph behaviors
Problem
Write the least-degree polynomial that crosses the \(x\text{-}\text{axis}\) at \(-2\) and \(3\) and passes through \((0,~6)\).
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What this problem is really about
Convert each graph behavior into the smallest multiplicity that produces it; a crossing needs an odd multiplicity, so least degree begins with simple factors. Attach an unknown overall scale and use the given point to determine it. A final check should confirm both crossings, the point, and that the factor exponents have the smallest possible sum.
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